How to tell the monotonicity from the implicit expression

Question

I have an equation as $f(x,y)=0$, and $f$ is very complicated that there is no way to rewrite it as $y=g(x)$ explicitly.

I would like to show that $y$ is increasing at $x$, is there any way I can show this from $f(x,y)=0$ instead of the classic ways such as to prove that $g'(x)>0$?

Many thanks for the answers!

Answer 1

From Wikipedia's article on implicit functions, if $R(x,y)=0$:

$$\frac{dy}{dx} = -\frac{\partial R/ \partial x}{\partial R / \partial y} = -\frac {R_x}{R_y}$$

This formula looks like it directly responds to your problem.